FurkanNar/Spatial_Context_Networks
Text Classification • Updated • 1
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classes | combined_text stringlengths 170 17.3k | scraped_at timestamp[ns]date 2026-02-26 15:45:26 2026-02-26 15:56:19 |
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733,754 | Visually stunning math concepts which are easy to explain | Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time. Do you know of any other concepts like these? | 1,670 | soft-question, recreational-mathematics, education, big-list, visualization | https://math.stackexchange.com/questions/733754/visually-stunning-math-concepts-which-are-easy-to-explain | I think if you look at this animation and think about it long enough, you'll understand: Why circles and right-angle triangles and angles are all related. Why sine is "opposite over hypotenuse" and so on. Why cosine is simply sine but offset by $\frac{\pi}{2}$ radians. | 1,173 | false | Q: Visually stunning math concepts which are easy to explain
Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time. Do you know of any oth... | 2026-02-26T15:45:26.800000 |
21,199 | Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio? | In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are able to plug in values for $\textrm{d}x$ and calculate a... | 1,334 | calculus, analysis, math-history, nonstandard-analysis | https://math.stackexchange.com/questions/21199/is-frac-textrmdy-textrmdx-not-a-ratio | Historically, when Leibniz conceived of the notation, $\frac{dy}{dx}$ was supposed to be a quotient: it was the quotient of the "infinitesimal change in $y$ produced by the change in $x$" divided by the "infinitesimal change in $x$". However, the formulation of calculus with infinitesimals in the usual setting of the r... | 1,608 | true | Q: Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?
In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because according to the formula $\textrm{d}y = f'(x)\textrm{d}x$ we are ab... | 2026-02-26T15:45:28.686000 |
379,927 | If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces? | So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her to saw another board into $3$ pieces? I don't understand what's wrong with this question. I think the student... | 1,102 | arithmetic, word-problem | https://math.stackexchange.com/questions/379927/if-it-took-10-minutes-to-saw-a-board-into-2-pieces-how-long-will-it-take-to-saw | Haha! The student probably has a more reasonable interpretation of the question. Of course, cutting one thing into two pieces requires only one cut! Cutting something into three pieces requires two cuts! ------------------------------- 0 cuts/1 piece/0 minutes ---------------|--------------- 1 cut/2 pieces/10 minutes -... | 922 | true | Q: If it took 10 minutes to saw a board into 2 pieces, how long will it take to saw another into 3 pieces?
So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long will it take for her t... | 2026-02-26T15:45:30.054000 |
71,874 | Can I use my powers for good? | I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that I am going to leave academia soon. I do enjoy teaching and research, but the alpha-maleness, massive egos and pressure to... | 919 | soft-question, career-development | https://math.stackexchange.com/questions/71874/can-i-use-my-powers-for-good | If you are in the US, there are several thousand institutions of higher learning, and at many of them there is very little "pressure to publish". At others, the "pressure to publish" can be met by publishing a textbook or some work of scholarship that does not require proofs of interesting (original) results. High scho... | 313 | false | Q: Can I use my powers for good?
I hesitate to ask this question, but I read a lot of the career advice from MathOverflow and math.stackexchange, and I couldn't find anything similar. Four years after the PhD, I am pretty sure that I am going to leave academia soon. I do enjoy teaching and research, but the alpha-male... | 2026-02-26T15:45:32.865000 |
12,906 | The staircase paradox, or why $\pi\ne4$ | What is wrong with this proof? Is $\pi=4?$ | 914 | geometry, analysis, convergence-divergence, pi, fake-proofs | https://math.stackexchange.com/questions/12906/the-staircase-paradox-or-why-pi-ne4 | This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is $2$, then take shorter and shorter stair-steps and the length is $2$ but your path approaches the diagonal. So $\sqrt{2}=2$. In both cases,... | 574 | true | Q: The staircase paradox, or why $\pi\ne4$
What is wrong with this proof? Is $\pi=4?$
A: This question is usually posed as the length of the diagonal of a unit square. You start going from one corner to the opposite one following the perimeter and observe the length is $2$, then take shorter and shorter stair-steps a... | 2026-02-26T15:45:35.011000 |
358,423 | A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals? | Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'm aware of uses quite a bit of commutative algebra and non-trivial theorems such as Krull's intersection theorem. It is ... | 895 | ring-theory, commutative-algebra, noetherian, krull-dimension, dimension-theory-algebra | https://math.stackexchange.com/questions/358423/a-proof-of-dimrt-dimr1-without-prime-ideals | Theorem. Let $R$ be a Noetherian commutative ring and let $T$ be an indeterminate. Then $$ \dim(R[T])=\dim(R)+1, $$ where $\dim$ denotes the Krull dimension, taken here via the Coquand–Lombardi (CL) first–order characterization described below. Proof. Write $d:=\dim(R)$. The proof splits into two inequalities. Preparat... | 1 | false | Q: A proof of $\dim(R[T])=\dim(R)+1$ without prime ideals?
Background. If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$, where $\dim$ denotes the Krull dimension. If $R$ is Noetherian, we have equality. Every proof of this fact I'm aware of uses quite a bit of commutative algebra and non-t... | 2026-02-26T15:45:37.981000 |
8,337 | Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem) | As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$ However, Euler was Euler and he gave other proofs. I believe many of you know some nice proofs of this, can you please share it with us? | 890 | sequences-and-series, fourier-analysis, big-list, faq, euler-sums | https://math.stackexchange.com/questions/8337/different-ways-to-prove-sum-k-1-infty-frac1k2-frac-pi26-the-b | OK, here's my favorite. I thought of this after reading a proof from the book "Proofs from the book" by Aigner & Ziegler, but later I found more or less the same proof as mine in a paper published a few years earlier by Josef Hofbauer. On Robin's list, the proof most similar to this is number 9 (EDIT: ...which is actua... | 402 | true | Q: Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem) $$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$ However, Euler was Euler and he gave other proofs... | 2026-02-26T15:45:40.179000 |
44,704 | How to study math to really understand it and have a healthy lifestyle with free time? | Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exercise, I've gained $30$ kg, I've spent countless hours studying at night, constantly had sl... | 866 | soft-question, advice | https://math.stackexchange.com/questions/44704/how-to-study-math-to-really-understand-it-and-have-a-healthy-lifestyle-with-free | In my view the central question that you should ask yourself is what is the end goal of your studies. As an example, American college life as depicted in film is hedonistic and certainly not centered on actual studies. Your example is the complete opposite - you describe yourself as an ascetic devoted to scholarship. M... | 290 | true | Q: How to study math to really understand it and have a healthy lifestyle with free time?
Here's my issue I faced; I worked really hard studying Math, so because of that, I started to realised that I understand things better. However, that comes at a big cost: In the last few years, I had practically zero physical exe... | 2026-02-26T15:45:41.808000 |
668 | What's an intuitive way to think about the determinant? | In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\times 2$ matrix by the formula. Our teacher showed us how to compute... | 848 | linear-algebra, matrices, determinant, intuition | https://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant | Your trouble with determinants is pretty common. They’re a hard thing to teach well, too, for two main reasons that I can see: the formulas you learn for computing them are messy and complicated, and there’s no “natural” way to interpret the value of the determinant, the way it’s easy to interpret the derivatives you d... | 518 | true | Q: What's an intuitive way to think about the determinant?
In my linear algebra class, we just talked about determinants. So far I’ve been understanding the material okay, but now I’m very confused. I get that when the determinant is zero, the matrix doesn’t have an inverse. I can find the determinant of a $2\time... | 2026-02-26T15:45:44.648000 |
216,343 | Does $\pi$ contain all possible number combinations? | $\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, time and manner of your death, and the answers to all the g... | 811 | elementary-number-theory, irrational-numbers, pi | https://math.stackexchange.com/questions/216343/does-pi-contain-all-possible-number-combinations | It is not true that an infinite, non-repeating decimal must contain ‘every possible number combination’. The decimal $0.011000111100000111111\dots$ is an easy counterexample. However, if the decimal expansion of $\pi$ contains every possible finite string of digits, which seems quite likely, then the rest of the statem... | 1,021 | true | Q: Does $\pi$ contain all possible number combinations?
$\pi$ Pi Pi is an infinite, nonrepeating $($sic$)$ decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of digits is the name of every person you will ever love, the date, ti... | 2026-02-26T15:45:47.214000 |
637,728 | Splitting a sandwich and not feeling deceived | This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and whispers whole day: "You will never solve me..." Please save me from this ... | 697 | game-theory, fair-division | https://math.stackexchange.com/questions/637728/splitting-a-sandwich-and-not-feeling-deceived | For more than two, the moving knife is a nice solution. Somebody takes a knife and moves it slowly across the sandwich. Any player may say "cut". At that moment, the sandwich is cut and the piece given to the one who said "cut". As he has said that is an acceptable piece, he believes he has at least $\frac 1n$ of the s... | 288 | false | Q: Splitting a sandwich and not feeling deceived
This is a problem that has haunted me for more than a decade. Not all the time - but from time to time, and always on windy or rainy days, it suddenly reappears in my mind, stares at me for half an hour to an hour, and then just grins at me, and whispers whole day: "You... | 2026-02-26T15:45:49.475000 |
952,466 | Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not? | Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ is connected. Question: Assume $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with th... | 664 | general-topology, metric-spaces, examples-counterexamples, connectedness | https://math.stackexchange.com/questions/952466/is-there-a-bijection-of-mathbbrn-with-itself-such-that-the-forward-map-is | 0 | false | Q: Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ i... | 2026-02-26T15:45:50.865000 | |
323,334 | What was the first bit of mathematics that made you realize that math is beautiful? (For children's book) | I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and found that I, too, lament the uninspiring quality of my elementary math education. I want to make a book that dis... | 659 | soft-question, education, big-list | https://math.stackexchange.com/questions/323334/what-was-the-first-bit-of-mathematics-that-made-you-realize-that-math-is-beautif | This wasn't the first, but it's definitely awesome: This is a proof of the Pythagorean theorem, and it uses no words! | 315 | false | Q: What was the first bit of mathematics that made you realize that math is beautiful? (For children's book)
I'm a children's book writer and illustrator, and I want to to create a book for young readers that exposes the beauty of mathematics. I recently read Paul Lockhart's essay "The Mathematician's Lament," and... | 2026-02-26T15:45:53.957000 |
1,681,993 | Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real? | So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}. $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting quadratic equation and assuming that a continued fraction that only contains... | 651 | complex-numbers, recursion, continued-fractions | https://math.stackexchange.com/questions/1681993/why-is-1-frac11-frac11-ldots-not-real | You're attempting to take a limit. $$x_{n+1} = 1-\frac{1}{x_n}$$ This recurrence actually never converges, from any real starting point. Indeed, $$x_2 = 1-\frac{1}{x_1}; \\ x_3 = 1-\frac{1}{1-1/x_1} = 1-\frac{x_1}{x_1-1} = \frac{1}{1-x_1}; \\ x_4 = x_1$$ So the sequence is periodic with period 3. Therefore it converges... | 558 | true | Q: Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?
So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}}. $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as $x = 1 + \dfrac{1}{x}$, solving the resulting quadratic equa... | 2026-02-26T15:45:56.209000 |
111,440 | Examples of patterns that eventually fail | Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of number of edges on a complete graph starts $0,1,3,6... | 643 | big-list, examples-counterexamples | https://math.stackexchange.com/questions/111440/examples-of-patterns-that-eventually-fail | I'll translate an entry in the blog Gaussianos ("Gaussians") about Polya's conjecture, titled: A BELIEF IS NOT A PROOF. We'll say a number is of even kind if in its prime factorization, an even number of primes appear. For example $6 = 2\cdot 3$ is a number of even kind. And we'll say a number is of odd kind if the num... | 436 | false | Q: Examples of patterns that eventually fail
Often, when I try to describe mathematics to the layman, I find myself struggling to convince them of the importance and consequence of "proof". I receive responses like: "surely if Collatz is true up to $20×2^{58}$, then it must always be true?"; and "the sequence of numbe... | 2026-02-26T15:45:58.460000 |
2,755 | Why can you turn clothing right-side-out? | My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily amused, and so am I). So he learned that you ca... | 640 | general-topology, algebraic-topology | https://math.stackexchange.com/questions/2755/why-can-you-turn-clothing-right-side-out | First, a warning. I suspect this response is likely not going to be immediately comprehensible. There is a formal set-up for your question, there are tools available to understand what's going on. They're not particularly light tools, but they exist and they're worthy of being mentioned. Before I write down the main th... | 292 | false | Q: Why can you turn clothing right-side-out?
My nephew was folding laundry, and turning the occasional shirt right-side-out. I showed him a "trick" where I turned it right-side-out by pulling the whole thing through a sleeve instead of the bottom or collar of the shirt. He thought it was really cool (kids are easily a... | 2026-02-26T15:46:01.636000 |
562,694 | Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$ | I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The approximate numeric value of the integral: $$I\approx8.372211626601275661625747121...$$ Neither Mathematica nor Maple cou... | 632 | calculus, integration, definite-integrals, contour-integration, closed-form | https://math.stackexchange.com/questions/562694/integral-int-11-frac1x-sqrt-frac1x1-x-ln-left-frac2-x22-x1 | I will transform the integral via a substitution, break it up into two pieces and recombine, perform an integration by parts, and perform another substitution to get an integral to which I know a closed form exists. From there, I use a method I know to attack the integral, but in an unusual way because of the 8th degre... | 1,076 | true | Q: Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \mathrm dx$
I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The app... | 2026-02-26T15:46:04.207000 |
11,669 | Mathematical difference between white and black notes in a piano | The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me. Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the distinction, then? Why not just have scales with $12$ notes? (apparen... | 601 | music-theory | https://math.stackexchange.com/questions/11669/mathematical-difference-between-white-and-black-notes-in-a-piano | The first thing you have to understand is that notes are not uniquely defined. Everything depends on what tuning you use. I'll assume we're talking about equal temperament here. In equal temperament, a half-step is the same as a frequency ratio of $\sqrt[12]{2}$; that way, twelve half-steps makes up an octave. Why twel... | 562 | true | Q: Mathematical difference between white and black notes in a piano
The division of the chromatic scale in $7$ natural notes (white keys in a piano) and $5$ accidental ones (black) seems a bit arbitrary to me. Apparently, adjacent notes in a piano (including white or black) are always separated by a semitone. Why the ... | 2026-02-26T15:46:06.423000 |
75,130 | How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$? | How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found out, that proving the above statement is enough for proving ... | 569 | calculus, limits, trigonometry, limits-without-lhopital | https://math.stackexchange.com/questions/75130/how-to-prove-that-lim-limits-x-to0-frac-sin-xx-1 | The area of $\triangle ABC$ is $\frac{1}{2}\sin(x)$. The area of the colored wedge is $\frac{1}{2}x$, and the area of $\triangle ABD$ is $\frac{1}{2}\tan(x)$. By inclusion, we get $$ \frac{1}{2}\tan(x)\ge\frac{1}{2}x\ge\frac{1}{2}\sin(x)\tag{1} $$ Dividing $(1)$ by $\frac{1}{2}\sin(x)$ and taking reciprocals, we get $$... | 675 | true | Q: How to prove that $\lim\limits_{x\to0}\frac{\sin x}x=1$?
How can one prove the statement $$\lim_{x\to 0}\frac{\sin x}x=1$$ without using the Taylor series of $\sin$, $\cos$ and $\tan$? Best would be a geometrical solution. This is homework. In my math class, we are about to prove that $\sin$ is continuous. We found... | 2026-02-26T15:46:08.599000 |
154 | Do complex numbers really exist? | Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that complex numbers are necessary and meani... | 546 | soft-question, complex-numbers, education, philosophy | https://math.stackexchange.com/questions/154/do-complex-numbers-really-exist | There are a few good answers to this question, depending on the audience. I've used all of these on occasion. A way to solve polynomials We came up with equations like $x - 5 = 0$, what is $x$?, and the naturals solved them (easily). Then we asked, "wait, what about $x + 5 = 0$?" So we invented negative numbers. Then w... | 378 | true | Q: Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious and intuitive meaning. What's the best way to explain to a non-mathematician that co... | 2026-02-26T15:46:10.545000 |
302,023 | Best Sets of Lecture Notes and Articles | Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting through books to find [the best source]". It strikes me now that while I love books (I really do), I often find that I lea... | 538 | self-learning, big-list, learning, online-resources | https://math.stackexchange.com/questions/302023/best-sets-of-lecture-notes-and-articles | In no particular order: Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615 Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf Representation theory resources: http://www.math.columbia.ed... | 72 | false | Q: Best Sets of Lecture Notes and Articles
Let me start by apologizing if there is another thread on math.se that subsumes this. I was updating my answer to the question here during which I made the claim that "I spend a lot of time sifting through books to find [the best source]". It strikes me now that while I love ... | 2026-02-26T15:46:13.822000 |
206,890 | "The Egg:" Bizarre behavior of the roots of a family of polynomials. | In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n$-dimensional vector space over $GF(x)$ (which I was using to determine the number ... | 529 | abstract-algebra, complex-analysis, algebraic-geometry, numerical-methods, recreational-mathematics | https://math.stackexchange.com/questions/206890/the-egg-bizarre-behavior-of-the-roots-of-a-family-of-polynomials | First, has anybody ever seen anything at all like this before? Yes, and in fact the interesting patterns that arise here are more than just a mathematical curiosity, they can be interpreted to have a physical context. Statistical Mechanics In a simple spin system, say the Ising model, a discrete set of points are arran... | 157 | false | Q: "The Egg:" Bizarre behavior of the roots of a family of polynomials.
In this MO post, I ran into the following family of polynomials: $$f_n(x)=\sum_{m=0}^{n}\prod_{k=0}^{m-1}\frac{x^n-x^k}{x^m-x^k}.$$ In the context of the post, $x$ was a prime number, and $f_n(x)$ counted the number of subspaces of an $n... | 2026-02-26T15:46:15.262000 |
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- Formats: CSV and JSON
- Typical fields:
question_id(int)question_title(string)question_body(string, cleaned of HTML)question_score(int)question_tags(string, comma-separated)question_url(string)answer_body(string, cleaned of HTML; empty if no answer)answer_score(int)answer_accepted(bool)combined_text(string, "Q:\n\n<body>\n\nA: <answer or placeholder>")</li> <li><code>scraped_at</code> (ISO 8601 timestamp)</li> </ul> </li> </ul> <p>Example CSV header:</p> <pre><code>question_id,question_title,question_body,question_score,question_tags,question_url,answer_body,answer_score,answer_accepted,combined_text,scraped_at </code></pre> <h2 class="relative group flex items-baseline"> <a id="license-and-attribution" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#license-and-attribution" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> License and attribution </span> </h2> <p>Content in this dataset is user-contributed content from Stack Exchange and is licensed under the Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0) license.</p> <p>When using or redistributing the dataset, include attribution. A suggested attribution statement:</p> <p>"Data from Mathematics Stack Exchange (math.stackexchange.com). Licensed under CC BY-SA 4.0."</p> <p>Also include a link to the original question pages (the <code>question_url</code> field) when practical.</p> <h2 class="relative group flex items-baseline"> <a id="how-the-dataset-was-generated" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#how-the-dataset-was-generated" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> How the dataset was generated </span> </h2> <p>The dataset was produced by a script that uses the official Stack Exchange API:</p> <ul> <li>Requests top-voted mathematics questions (configurable count)</li> <li>For each question, fetches the top answer (if any)</li> <li>Cleans HTML from bodies (HTML entities unescaped and tags removed), collapses whitespace</li> <li>Produces both CSV and JSON exports</li> <li>Respects rate limits and inserts polite delays</li> </ul> <p>Notes:</p> <ul> <li>The generator script is not included in this repository. If you want to include it, add the script file (for example <code>scripts/generate_math_se_dataset.py</code>) and reference it here.</li> <li>For reproducibility, include the script, the exact arguments used, and the timestamp of generation.</li> </ul> <h2 class="relative group flex items-baseline"> <a id="regenerating-the-dataset" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#regenerating-the-dataset" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Regenerating the dataset </span> </h2> <p>Prerequisites:</p> <ul> <li>Python 3.8+ (or compatible)</li> <li><code>requests</code> (for API calls)</li> <li>Optional: a Stack Exchange API key (recommended for higher rate limits)</li> </ul> <p>Important considerations:</p> <ul> <li>Respect Stack Exchange API rate limits; use an API key if you will fetch many pages.</li> <li>The script should include backoff/retry and delays between requests; do not parallelize requests without careful rate-limit handling.</li> <li>If exporting large datasets, consider paging and storing intermediate results.</li> </ul> <h2 class="relative group flex items-baseline"> <a id="data-quality--known-limitations" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#data-quality--known-limitations" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Data quality & known limitations </span> </h2> <ul> <li>Bodies are cleaned of HTML but some formatting (math markup, code blocks, LaTeX) may be simplified or lost; consider preserving LaTeX if you need formula fidelity.</li> <li>Only the top answer (by score) is captured — accepted vs. highest-scored may differ; both are recorded if available.</li> <li>Potential biases:<ul> <li>Questions with higher votes and English-language questions are overrepresented if using a top-voted query.</li> <li>Highly specialized or duplicate content may appear.</li> </ul> </li> <li>Empty <code>answer_body</code> means no answer was available at scraping time.</li> <li>Some fields may contain non-ASCII characters; files are exported with UTF-8 encoding.</li> </ul> <h2 class="relative group flex items-baseline"> <a id="suggested-preprocessing-for-ml" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#suggested-preprocessing-for-ml" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Suggested preprocessing for ML </span> </h2> <ul> <li>Deduplicate entries by <code>question_id</code>.</li> <li>Normalize or preserve LaTeX as needed (consider using a LaTeX-aware tokenizer when training models that must understand formulas).</li> <li>Truncate or chunk very long texts (both question and answer) when fitting into model context windows.</li> <li>Optionally filter by tags for domain-specific datasets (e.g., <code>algebra</code>, <code>calculus</code>, <code>probability</code>).</li> </ul> <h2 class="relative group flex items-baseline"> <a id="use-cases" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#use-cases" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Use cases </span> </h2> <ul> <li>Training and evaluating question-answering systems and math tutoring chatbots</li> <li>Fine-tuning language models on math Q&A patterns</li> <li>Research on problem-solving strategies and reasoning in mathematics</li> <li>Building retrieval-augmented systems with real Q&A examples</li> </ul> <h2 class="relative group flex items-baseline"> <a id="privacy-ethics-and-policy" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#privacy-ethics-and-policy" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Privacy, ethics, and policy </span> </h2> <ul> <li>The dataset contains user-contributed content under CC BY-SA; comply with the license and provide attribution.</li> <li>Do not claim the content as your own; include proper attribution and linkbacks where practical.</li> <li>Avoid publishing personally identifiable information beyond what is already publicly available on Stack Exchange.</li> </ul> <h2 class="relative group flex items-baseline"> <a id="citation" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#citation" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Citation </span> </h2> <p>If you use this dataset in publications, we suggest citing it as:</p> <p>Furkan Nar or <a href="https://github.com/TheOfficialFurkanNar/StackMathematics" rel="nofollow">https://github.com/TheOfficialFurkanNar/StackMathematics</a>. (2026). Mathematics Stack Exchange Q&A Dataset. Retrieved from <a href="https://github.com/TheOfficialFurkanNar/StackMathematics" rel="nofollow">https://github.com/TheOfficialFurkanNar/StackMathematics</a> (data originally from mathematics.stackexchange.com; licensed CC BY-SA 4.0)</p> <h2 class="relative group flex items-baseline"> <a id="contact" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#contact" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Contact </span> </h2> <p>Contact via: <a href="mailto:furkannar168@hotmail.com" rel="nofollow">furkannar168@hotmail.com</a> For questions about this dataset or generation script, open an issue in this repository or contact the maintainer.</p> <h2 class="relative group flex items-baseline"> <a id="community--support" class="block pr-1.5 text-lg md:absolute md:p-1.5 md:opacity-0 md:group-hover:opacity-100 md:right-full" href="#community--support" > <span class="header-link"><!--[--><svg class="text-gray-500 hover:text-black dark:hover:text-gray-200 w-4" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" role="img" width="1em" height="1em" preserveAspectRatio="xMidYMid meet" viewBox="0 0 256 256"><path d="M167.594 88.393a8.001 8.001 0 0 1 0 11.314l-67.882 67.882a8 8 0 1 1-11.314-11.315l67.882-67.881a8.003 8.003 0 0 1 11.314 0zm-28.287 84.86l-28.284 28.284a40 40 0 0 1-56.567-56.567l28.284-28.284a8 8 0 0 0-11.315-11.315l-28.284 28.284a56 56 0 0 0 79.196 79.197l28.285-28.285a8 8 0 1 0-11.315-11.314zM212.852 43.14a56.002 56.002 0 0 0-79.196 0l-28.284 28.284a8 8 0 1 0 11.314 11.314l28.284-28.284a40 40 0 0 1 56.568 56.567l-28.285 28.285a8 8 0 0 0 11.315 11.314l28.284-28.284a56.065 56.065 0 0 0 0-79.196z" fill="currentColor"></path></svg><!--]--></span> </a> <span> Community & Support </span> </h2> <p>Join the community to ask questions, discuss the dataset, and get help:</p> <ul> <li>Discord: <a href="https://discord.gg/dcjh8S5H" rel="nofollow">https://discord.gg/dcjh8S5H</a></li> </ul>
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